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\begin{center}\bf
Finger Exercises --- Due Thursday January 29, 1998
\end{center}

\begin{quote}\em
Please prepare answers for these exercises, to be turned in.  Your
grade will be based on how many questions you made a reasonably strong
effort to answer, {\bf not} how many questions you get right or
partially right.  There is no partial credit, but the grader may give
you additional written feedback on each question you attempt.

Feel free to discuss these exercises with each other and with me.  You
will get the most benefit from the grader's remarks, however, if the
work you turn in is your own.

\rm

This hw, like most material for this course, is posted on the World
Wide Web at URL \www{}; you can cut and paste data directly out of the
\TeX{} file if you wish.  Look for something under week2.

The first two questions are ``reviews'' of your previous work in
Statistics and of your common sense, calculus skill, etc.  The
remaining questions are about ANOVA.
\end{quote}

\begin{enumerate}

\item {\em Box, Hunter and Hunter, p. 155.} Every 90 minutes, routine
  readings are made of the level of asbestos fiber present in the air
  at an industrial plant.  A salesman claimed that spraying with a
  chemical, S-424, could be beneficial.  Arrangements were made for a
  comparative trial in the plant itself.  Four consecutive readings,
  the first two without S-424 and the second two with S-424, were as
  follows:
\[
        8 ~~~~~~~~~~~
        6 ~~~~~~~~~~~
        3 ~~~~~~~~~~~
        4 
\]
In light of past data collected without the chemical spray, given
below, do you think the additive works?
\begin{verbatim}
                  Asbestos Levels (112 consecutive readings)
        Dec  6    9 10  9  8  9  8  8  8  7  6  9 10 11  9 10 11
        Dec  7   11 11 11 10 11 12 13 12 13 12 14 15 14 12 13 13 
        Dec  8   12 13 13 13 13 13 10  8  9  8  6  7  7  6  5  6
        Dec  9    5  6  4  5  4  4  2  4  5  4  5  6  5  5  6  5
        Dec 10    6  7  8  8  8  7  9 10  9 10  9  8  9  8  7  7
        Dec 11    8  7  7  7  8  8  8  8  7  6  5  6  5  6  7  6
        Dec 12    6  5  6  6  5  4  3  4  5  5  6  5  6  7  6  5
\end{verbatim}
Explain and justify your answer, using both statistical and
nonstatistical means.  Explicitly state {\em all} assumptions you are
making about the problem.  Do not worry about whether or not your answer
involves regression, ANOVA, or any other particular method; just do
the right things.

\item {\em Box, Hunter and Hunter, p. 158.} An agricultural engineer
  is trying to develop a special piece of equipment for processing a
  crop immediately after harvesting.  Two configurations of this
  equipment are to be compared.  Twenty runs are to be performed
  during a period of five days.  Each run involves setting up the
  equipment in either configuration I or configuration II, harvesting
  a given amount of crop, processing it on the equipment, obtaining a
  quantitative measure of performance, and cleaning the equipment.
  Since there is only one piece of equipment, the tests must be done
  one at a time. The engineer has asked you to consult on this
  problem.
\begin{enumerate}
\item What questions would you ask him?
\item What advice might you give him about planning the experiment?
\item The engineer believes that a most accurate experiment requires
  an equal number of runs to be performed with each configuration.
  What assumptions would make it possible to demonstrate in a
  quantitative way that this is true?
\end{enumerate}

\item In the ``grand mean plus effects'' model of lecture from
Thursday Jan 22, suppose that we define $\mu_. = \sum_{i=1}^k w_i
\mu_i$ in terms of the cell means $\mu_i$, where $w_i$ are positive
constants such that $\sum_{i=1}^k w_i = 1$.  If we now define the
treatment effects to be $\alpha_i = \mu_i - \mu$, show that
\[
	\sum_{i=1}^k w_i\alpha_i = 0
\]

\ \\

\item[{\bf Please}] {\bf do some analysis for both questions 4 and 5,
but only write up one of them.}

\item {\bf Questionnaire color.} [Neter, Wasserman and Kutner, 1990,
pp. 562--563.] In an experiment to investigate the effect of color of
paper (blue, green, orange) on response rates for questionnaires
distributed by the ``windshield method'' in supermarket parking lots,
15 representative supermarket parking lots were chosen in a
metropolitan area, and each color was assigned at random to five of
the lots.  The response rates, in percents, are listed in the table
below.
\begin{verbatim}
  BLUE 28 26 31 27 35
 GREEN 34 29 25 31 29
ORANGE 31 25 27 29 28
\end{verbatim}

\begin{enumerate}
\item Is this an experiment or an observational study?  Explain.

\item Fit the one-way analysis of variance model; test the hypothesis
$H_0:$ the response rates vary depending on the color.

\item Do the data look like they satisfy the assumptions of the ANOVA
/ linear regression model?  Inspect boxplots at each of the factor
levels, fitted, residual and QQ plots, and use whatever other evidence
you can think of to answer this question.

\item If you found evidence of unequal variances in the cells,
nonnormality of residuals, or other problems, explain the problem(s)
and try to fix them as best you can.  If neccesary, re-run the ANOVA
problem with the fixed dataset.

\item Obtain naive (that is, not adjusted for multiple comparisons)
confidence intervals for each of the following:
\begin{itemize}
\item The cell means in the cell means model $y_{ij} = \mu_i +
\epsilon_{ij}$;
\item The treatment (factor level) effects in the grand mean plus
effects model $y_{ij} = \mu + \alpha_i + \epsilon_{ij}$, with the
constraint that $\sum_i \alpha_i=0$.
\end{itemize}

\item When informed of the findings, an executive said, ``See?  I was
right all along.  We might as well print the questionnaires on plain
white paper, which is cheaper!''  Does this conclusion follow from the
findings of the study?  Discuss.

\end{enumerate}

\item {\bf Rehabilitation therapy.} [Neter, Wasserman and Kutner,
1990, pp. 562--563.]  A rehabilitation center researcher was
interested in examining the relationship between physical fitness
prior to persons undergoing corrective knee surgery and time required
in physical therapy until successful rehabilitation.  Patient records
in the rehabilitation center were examined, and 24 make subjects
ranging in age from 18 to 34 years who had undergone similar
corrective knee surgery during the past year were selected for the
study.  The number of days required for successful completion of
physical therapy, and the prior physical fitness status, are listed in
the table below.
\begin{verbatim}
Fitness Score Nobs  1  2  3  4  5  6  7  8  9 10 
  BELOW    83    8 29 42 38 40 43 40 30 42      
AVERAGE   100   10 30 35 39 28 31 31 29 35 29 33
  ABOVE   121    6 26 32 21 20 23 22 
\end{verbatim}
In the table, ``Score'' is the mean score of men in each fitness
group, on a standard medical fitness checklist (scores could range
from 0 to 150), and ``Nobs'' is the number of observations in each
fitness group.
\begin{enumerate}
\item Is this an experiment or an observational study?  Explain.

\item Fit the one-way analysis of variance model; test the hypothesis
$H_0:$ the recovery (days needed in physical therapy) times vary
depending on the prior fitness of the patient.

\item Do the data look like they satisfy the assumptions of the ANOVA
/ linear regression model?  Inspect boxplots at each of the factor
levels, fitted, residual and QQ plots, and use whatever other evidence
you can think of to answer this question.

\item If you found evidence of unequal variances in the cells,
nonnormality of residuals, or other problems, explain the problem(s)
and try to fix them as best you can.  If neccesary, re-run the ANOVA
problem with the fixed dataset.

\item Estimate a 99\% confidence interval for the mean recovery time
(days in physical therapy) needed by someone of average fitness prior
to the surgery.

\item Obtain confidence intervals for $D_1=\mu_2 - \mu_3$ and $D_2=\mu_1
- \mu_2$, using each of the three methods:
\begin{itemize}
\item Bonferroni
\item Scheff\'e
\item Tukey
\end{itemize}
Which method gives the best (shortest) intervals, in this case?

\item Now suppose you also want to compute an interval for $D_3=\mu_1
- \mu_3$.  Which of the three procedures has to be recomputed?  Why?
Give the new interval(s) for any procedure(s) that need recomputing.

\item The scores 83, 100 and 121 were the average fitness scores in
each fitness group.  Can you say anything reasonable about the
relationship between the numerical fitness score of a patient and the
recovery time needed?  Discuss.

\end{enumerate}

\end{enumerate}


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